Roberts, Linear stability analysis of the figure-eight orbit in the three-body problem,Ergodic Theory Dynam. Systems27 Tanikawa, On the symmetric collinear four-body problem,Publication of the Astronomical Society of Japan56 orbits for nine binaries and one linear solution, Differential Equations98 Sweatman, The symmetrical one-dimensional Newtonian four-body problem: Sweatman, A family of symmetrical Schubart-like interplay orbits and their stability in the one-dimensional four-body problem,Celestial Mech.

Regularization of simultaneous binary collisions and solutions with singularity in the collinear four-body problem. BolotinPiero Negrini. A restricted four-body model for the dynamics near the Lagrangian points of the Sun-Jupiter system.

LacombaMario Medina. Oscillatory motions in the rectangular four body problem. Shiqing ZhangQing Zhou. New periodic solutions for three or four identical vortices on a plane and a sphere. Conference Publications, Special: Positive solutions of Kirchhoff type problem with singular and critical nonlinearities in dimension four. CabralZhihong Xia.

Subharmonic solutions in the restricted three-body problem. An existence proof of a symmetric periodic orbit in the octahedral six-body problem. Some remarks on symmetric periodic orbits in the restricted three-body problem.

Tiancheng Orbits for nine binaries and one linear solutionDuokui Yan. Variational properties and linear stabilities of spatial isosceles orbits in the equal-mass three-body problem. Horseshoe periodic orbits with one symmetry in the general planar three-body problem.

American Institute of Mathematical Sciences. Previous Article Solitary waves in critical Abelian gauge theories. Breather continuation from infinity in nonlinear oscillator chains. The Schubart-like orbits in the collinear four-body problem are similar to those discovered numerically by Schubart[12] in the collinear three-body problem.

Schubart-like orbits are periodic solutions with exactly two binary collisions and one simultaneous binary collision per period. The proof of the existence of these orbits given in this paper is based on the direct method of Calculus of Variations. We exploit the variational structure of the problem and show that the minimizers of the Lagrangian action functional in a suitably chosen space have the desired properties.

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Kepler's first and third laws. The Appendix gives a brief derivation of Kepler's first and third laws, starting from constant energy and constant angular momentum. Calculate, using Kepler's third law, the time needed for a spacecraft to travel from Earth to Mars, stay on Mars a while, and then return to Earth. Assume circular orbits for both Earth and Mars and a spacecraft orbit that just touches the two planet orbits. Round off all periods to simple fractions of a year.

Sketch the planetary positions at the time of launch and at the time of arrival at Mars, together with the spacecraft orbit from Earth to Mars, and again for the return trip.

In and for a few years thereafter, humans visited the Moon. The next major goal for human travel is Mars, perhaps by the year Even now, Mars is being explored intensively, by computer-directed spacecraft, because of indirect and debatable evidence that some form of primitive life may have occurred on Mars long ago.

To send humans to Mars is technically much more complicated. A massive spacecraft will be needed to assure human survival.

Its launch into an orbit to Mars will need the most powerful rockets available. Given likely rocket power, even twenty years from now, an orbit from Earth to Mars will have to be chosen to minimize the required energy. With the approximation that the orbits of Earth and Mars are circles, which is adequate for our purposes, the minimum-energy orbit is an ellipse practically a circle that just touches the orbits of Earth and Mars. To visit and stop at Mars, the orbit will consist of two half ellipses.

The spacecraft will be launched from Earth, perform half an orbit in traveling from Earth to Mars, land on Mars, later be launched from Mars, perform half an orbit, and land on Earth. It must stay on Mars long enough so that the Earth will be in the right position for the space craft to meet Earth. Now the period of Mars' orbit, again from Kepler's third law, is 2 years. While the explorers are on Mars, Earth must progress by half a full circle relative to Mars.

The difficulty of this trip resides not only in the engineering. They will be exposed to cosmic rays and energetic particles from solar flares. So far, we do not know how to assure the health of astronauts in space for so long a time. The experiences in the Russian space station Mir have begun to solve this question. The explorers will also be exposed to cosmic and solar particles while on the surface of Mars, but they may spend most of their time on Mars underground. For space probes without humans, it is no longer necessary to use the minimum-energy orbit.

Moreover, if there is enough time, space craft may obtain extra needed energy by suitably passing near any convenient planet see problem 2. This is an excellent problem for students to solve in small groups. The professor needs to walk among the groups, check their progress especially by checking their diagrams , watch for preconceptions such as item 3 below, and ask questions that will get the groups to progress when stuck.

Afterwards, or possibly part way through, the appointed reporter from different groups should be asked to report their results. Insist on students drawing the diagrams.

Diagrams focus the human mind remarkably well. Depending on class preparation, students should review, again in small groups, the physics leading to Kepler's third law for circular orbits around the Sun. Conventionally, the right side is set equal to unity upon expressing r in the unit of the semi-major axis of Earth's orbit, 1 Astronomical Unit A.

The derivation of Kepler's third law for elliptical orbits see Appendix does not provide any useful additional physical insight. Watch for a preconception: When they are asked to draw an elliptical orbit connecting Earth and Mars orbits, many students will draw an ellipse like this: If they do, ask them to figure out why the sketch is wrong.

Sun not at focus. Do not allow your students to calculate a semi-major axis or the number of seconds in a year accurate to 3 or 4 decimal places, because the approximations already made such as taking Mars' orbit to be circular cause errors even in the second decimal.

A sense of awareness of possible errors must be reflected in the number of decimals one uses. How do we know that 1 A. We can measure the distance to a planet like Mars or Mercury by radar: Historically, the distance of Mars was first derived by triangulation.

We know that same distance expressed in A. The asteroids, with sizes of our Moon or less, have low surface gravity and are easier to land on than is Mars.

Some asteroids may provide opportunities for mining. To simplify this problem, ignore the motion of Ceres during the time while the explorers are on Ceres. The orbital periods of the space craft and Ceres are about 2. Ignoring Ceres' motion means that Earth must be in the same place at launch and arrival, so the whole trip takes the minimum number of whole years, that is 3 years.

Therefore, the humans must spend 0. It is now possible to compute the problem in the next approximation: Find the mass of the Earth, given the lunar distance 4x10 5 km and lunar orbital period as seen by a very distant observer of 27 days. Energy equation for orbits around the Sun. The Appendix gives a short derivation starting from constant angular momentum and constant total energy.

Evaluate the difference in v 2. Compare to the value of v 2 needed to escape the Earth's gravity. Which takes less energy, leaving the Earth's gravity or changing from circular to elliptical orbit around the Sun? The space craft taking humans to Mars and back to Earth on a minimum-energy orbit tangent to the orbits of Earth and Mars, both assumed circular must use rocket propulsion to escape Earth's gravity, then to accelerate to acquire the minimum-energy orbit to Mars, later to accelerate to match the speed of Mars, and finally to reduce the speed of falling onto Mars; still later, rocket propulsion is again needed to escape Mars' gravity, to slow down to acquire the minimum-energy orbit to Earth, to slow down to match the speed of Earth, and finally to reduce the speed of falling to Earth.

This problem shows that all of these maneuvers require significant energies. They are compared to the energy needed to escape Earth's gravity because most students have seen pictures of major rocket launchings and have at least an intuitive feeling that this energy is quite large.

The energy needed for the space craft to escape the Earth's gravity is less than the energy needed, after escaping from Earth's gravity, to place the craft into the correct orbit around the Sun. The energies needed to change orbit are quite comparable to the energies needed to escape Earth's gravity.

The energies are near the limits of technical capability if the spacecraft is heavy enough to carry humans safely to Mars and back. The use of v 2 has in effect focused on the energies needed by the unfueled space craft occupied by humans.

Additional energy is needed for fuel, which is reduced every time the rockets are fired, and possibly discardable rocket stages. For instance, the spacecraft Ulysses reached Jupiter on a much shorter orbit.

It then gained energy while flying past Jupiter see problem 2. The energy equation is an appropriate topic for lecture. However, then the students can solve the problem in groups, with different students serving as recorders and reporters than for problem 1.

If one group reports some part of the problem incorrectly, let other student groups recognize the mistake. The professor need only make sure that the incorrect group figures out what they did not understand correctly.

The energy equation simply means: The derivation in the Appendix merely identifies the constant. Students may be tempted to evaluate v final -v initial during the time the rocket operates and then square that difference.

They will get different answers because they are implicitly placing themselves into different frames of reference.

All the energies of the spacecraft and planet orbits are evaluated in one frame, stationary with respect to some distant star. Check that your class realizes: Rocket power is used briefly to provide the necessary energy, but then the spacecraft moves without effort along its new Keplerian orbit, until the rockets must again be used briefly upon reaching Mars.

Check that your class understands the minimum-energy orbit qualitatively: Once free of Earth's gravity, the spacecraft is in a circular orbit around the Sun. The spacecraft must be accelerated so that it "overshoots" Earth's orbit to reach further out. Once at Mars, it must again be accelerated to achieve a circular orbit at the distance of Mars.

You can check that the class understands this: Ask for equivalent arguments about the decelerations on the way back from Mars to Earth. Once again, students should work out this problem using no more than two significant digits.

The differences in v 2 computed here are very sensitive to the approximations made, especially the approximation that Mars moves in a circular orbit. Two-body problem, for circular orbits. Doppler shift only for interpretation. Assume that the star moves because it is pulled by a planet orbiting in a circle around that star. Assume that our line of sight to the star is parallel to the plane of the planet orbit. Check that your equations work for Jupiter: Evaluate M p and r, in these units, for the star 47 UMa: The Sun is a fairly normal star.

Therefore, most astronomers expect that there are other stars with planets orbiting around them. But it has been difficult to detect planets. They cannot be photographed because they are too faint compared to the immediately adjacent star. The contrast is smaller in the infrared. With improving infrared technology, perhaps planets can be photographed in the infrared within a few years. Since , orbiting planets can be detected by the corresponding motion of the star action and reaction.